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Theorem nfcnv 5889
Description: Bound-variable hypothesis builder for converse relation. (Contributed by NM, 31-Jan-2004.) (Revised by Mario Carneiro, 15-Oct-2016.)
Hypothesis
Ref Expression
nfcnv.1 𝑥𝐴
Assertion
Ref Expression
nfcnv 𝑥𝐴

Proof of Theorem nfcnv
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cnv 5693 . 2 𝐴 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
2 nfcv 2905 . . . 4 𝑥𝑧
3 nfcnv.1 . . . 4 𝑥𝐴
4 nfcv 2905 . . . 4 𝑥𝑦
52, 3, 4nfbr 5190 . . 3 𝑥 𝑧𝐴𝑦
65nfopab 5212 . 2 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴𝑦}
71, 6nfcxfr 2903 1 𝑥𝐴
Colors of variables: wff setvar class
Syntax hints:  wnfc 2890   class class class wbr 5143  {copab 5205  ccnv 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-cnv 5693
This theorem is referenced by:  nfrn  5963  nfpred  6326  nffun  6589  nff1  6802  nfsup  9491  nfinf  9522  gsumcom2  19993  ptbasfi  23589  mbfposr  25687  itg1climres  25749  funcnvmpt  32677  nfwsuc  35819  aomclem8  43073  rfcnpre1  45024  rfcnpre2  45036  smfpimcc  46823
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